# Spatial disease mapping using Directed Acyclic Graph Auto-Regressive (DAGAR) models

## TL;DR

This paper introduces DAGAR, a new spatial modeling approach using directed acyclic graphs that ensures positive definiteness, improves interpretability, and enhances the accuracy of disease mapping compared to traditional models.

## Contribution

The paper proposes a novel DAG-based precision matrix model for spatial effects that guarantees positive definiteness and improves interpretability over existing methods.

## Key findings

- DAGAR outperforms CAR models under modest spatial correlation.
- The model provides accurate recovery of latent spatial effects.
- It demonstrates robustness to DAG ordering choices.

## Abstract

Hierarchical models for regionally aggregated disease incidence data commonly involve region specific latent random effects that are modeled jointly as having a multivariate Gaussian distribution. The covariance or precision matrix incorporates the spatial dependence between the regions. Common choices for the precision matrix include the widely used ICAR model, which is singular, and its nonsingular extension which lacks interpretability. We propose a new parametric model for the precision matrix based on a directed acyclic graph (DAG) representation of the spatial dependence. Our model guarantees positive definiteness and, hence, in addition to being a valid prior for regional spatially correlated random effects, can also directly model the outcome from dependent data like images and networks. Theoretical results establish a link between the parameters in our model and the variance and covariances of the random effects. Substantive simulation studies demonstrate that the improved interpretability of our model reaps benefits in terms of accurately recovering the latent spatial random effects as well as for inference on the spatial covariance parameters. Under modest spatial correlation, our model far outperforms the CAR models, while the performances are similar when the spatial correlation is strong. We also assess sensitivity to the choice of the ordering in the DAG construction using theoretical and empirical results which testify to the robustness of our model. We also present a large-scale public health application demonstrating the competitive performance of the model.